Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Let $C,D$ be 2-categories and $F,G:C\to D$ be functors. An icon $\alpha:F\to G$ consists of the following:
If $D$ is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-cell components are identities. In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of $D$. The name “icon” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.
Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)
Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.
Icons are also used to construct distributors in the context of enriched bicategories.
A bicategory can be viewed as a pseudo double category whose tight-cells are trivial. An icon is then precisely a transformation of oplax functors of pseudo double categories. See Paré for details.
An early observation that restricting to icons allows one to form a bicategory of bicategories and lax functors appears in:
The terminology “ICON” was introduced in:
Stephen Lack, Icons. Appl. Categ. Structures. Volume 18, Issue 3 (2010), pp 289–307. arxiv:0711.4657
Richard Garner, Mike Shulman, Section 3.8 of Enriched categories as a free cocompletion, Adv. Math. 289 (2016), 1–94
Robert Paré. Composition of modules for lax functors. Theory and Applications of Categories 27.16 (2013): 393-444.
Niles Johnson, Donald Yau, Section 4.6 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)
Last revised on October 16, 2023 at 09:34:23. See the history of this page for a list of all contributions to it.